Asymptotic behavior of solutions to half-linear \(q\)-difference equations. (English) Zbl 1208.39009

Řehák, Pavel

Abstr. Appl. Anal. 2011, Article ID 986343, 12 p. (2011).

Summary: We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linear \(q\)-difference equation \(D_q(\Phi(D_qy(t)))+p(t)\Phi(y(qt))=0\), \(t\in\{q^k:k\in\mathbb N_0\}\) with \(q>1\), \(\Phi(u)=|u|^{\alpha-1}\text{sgn\,}u\) with \(\alpha>1\), to behave like \(q\)-regularly varying or \(q\)-rapidly varying or \(q\)-regularly bounded functions (that is, the functions \(y\), for which a special limit behavior of \(q(qt)/y(t)\) as \(t\to\infty\) is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

MSC:

39A13	Difference equations, scaling (\(q\)-differences)
39A22	Growth, boundedness, comparison of solutions to difference equations

Keywords:

positive solutions; half-linear \(q\)-difference equation; asymptotic behavior

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